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Shifting definition (定義ずらし in Japanese) is a method to generate a googological system introduced by a Japanese Googology Wiki user Jason.[1] It is regarded as a generalisation of side nesting by its creator mrna.


Feature

Similar to side nesting, a \(2\)-ary function symbol \(+\) in a notation defined by shifting definition does not necessarily work as the addition. Namely, even if a valid expression \(a\) corresponds to a countable ordinal \(\alpha\) and \(a + a\) is also a valid expression, \(a + a\) does not necessarily corresponds to \(\alpha + \alpha\).

Another specific feature of shifting definition is that it is intended to be a correspondence which assigns a new defining formula to a defining formula of ordinal functions in a certain class. In order to formalise this strategy, we need to encode defining formulae in set theory in some way. Although the method has not been formalised yet, Jason intends to encode definining formulae into ordinals. There are two examples of unformalised googological systems which are intended to be justified by shifting definition through the encoding of defining formulae into ordinals. We will explain them in the next section.


Example

Since definition shifting is considered as a generalisation of side nesting, the notations in the articles of side nesting, which has not been formalised yet, are examples of notations associated to ordinal functions defined by shifting definition. Three other examples are given by Jason:

Although neither of them has been fully defined yet, they are expected to be significantly strong if they will be appropriately formalised.


δOCF

Main article: δOCF

δOCF is the first system defined by shifting definition, and is intended to perform as an ordinal function associated to an OCF through the method.[2][3][4]


δφ

Main article: δφ

δφ is the second system defined by shifting definition, and is intended to perform as an ordinal function associated to Veblen function through shifting definition.[5][6]


ε function

Main article: ε function

ε function is the third system defined by shifting definition, and it intend to perform as a notation inclusing shifting definition and higher system called meta-shifting definition, meta-meta-shifting definition, and so on.[7][8][9][10]


References

  1. The user page of Jason in Japanese Googology Wiki.
  2. Jason, δ関数の展開ルール, Google Drive.
  3. Jason, δ解説 もっと詳しく, Google Document.
  4. Jason, グラハム数〜legend of δ〜, twiiter.
  5. Jason, δφ対応表, Google Document.
  6. Jason, δφのいろいろ, Japanese Googology WIki user blog.
  7. Jason, ε関数定義試作, Google Document. (Trial to formalise the notation.)
  8. Jason, ε関数成長記録, Google Document. (Comparison to other notations.)
  9. Jason, ε関数 ver ε.0.1.0, Google Document. (The current version.)
  10. Jason, ε関数の解説を試みる, Japanese Googology Wiki user blog. (Trial to explain ε function)


See Also

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